Haldane's formula in Cannings models: The case of moderately strong selection

TL;DR

This paper proves Haldane's formula for the fixation probability of beneficial mutants in certain Cannings models under moderately strong selection, extending understanding of genetic drift and selection in large populations.

Contribution

It establishes Haldane's formula for a class of Cannings models with moderately strong selection, using coupling with Galton-Watson processes.

Findings

Haldane's formula holds for large populations under specified conditions.

The model allows for reproduction mechanisms coupled with Galton-Watson processes.

The fixation probability scales as 2s_N/ρ^2 in the considered regime.

Abstract

For a class of Cannings models we prove Haldane's formula, $\pi(s_N) \sim \frac{2s_N}{\rho^2}$, for the fixation probability of a single beneficial mutant in the limit of large population size $N$ and in the regime of moderately strong selection, i.e. for $s_N \sim N^{-b}$ and $0< b<1/2$. Here, $s_N$ is the selective advantage of an individual carrying the beneficial type, and $\rho^2$ is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele's frequency process with slightly supercritical Galton-Watson processes in the early phase of fixation.